# SAT Math : Inequalities

## Example Questions

### Example Question #2 : How To Find The Solution To An Inequality With Addition

Solve for .

Explanation:

Absolute value problems always have two sides: one positive and one negative.

First, take the problem as is and drop the absolute value signs for the positive side: z – 3 ≥ 5. When the original inequality is multiplied by –1 we get z – 3 ≤ –5.

Solve each inequality separately to get z ≤ –2 or z ≥ 8 (the inequality sign flips when multiplying or dividing by a negative number).

We can verify the solution by substituting in 0 for z to see if we get a true or false statement. Since –3 ≥ 5 is always false we know we want the two outside inequalities, rather than their intersection.

### Example Question #5 : How To Find The Solution To An Inequality With Addition

What values of  make the statement  true?

Explanation:

First, solve the inequality :

Since we are dealing with absolute value,  must also be true; therefore:

### Example Question #21 : Inequalities

Solve:

Explanation:

To solve , isolate .

Divide by three on both sides.

### Example Question #22 : Inequalities

Solve for .

Explanation:

We want to isolate the variable on one side and numbers on another side. Treat like a normal equation.

Subtract  on both sides.

Divide  on both sides. Remember to flip the sign.

### Example Question #23 : Inequalities

Solve for .

Explanation:

We want to isolate the variable on one side and numbers on another side. Treat like a normal equation.

Subtract  on both sides.

### Example Question #24 : Inequalities

Solve for .

Explanation:

We want to isolate the variable on one side and numbers on another side. Treat like a normal equation.

We need to set-up two equations since its absolute value.

Subtract  on both sides.

Divide  on both sides which flips the sign.

Subtract  on both sides.

Since we have the 's being either greater than or less than the values, we can combine them to get .

### Example Question #25 : Inequalities

Solve for .

Explanation:

We want to isolate the variable on one side and numbers on another side. Treat like a normal equation.

We need to set-up two equations since it's absolute value.

Subtract  on both sides.

Divide  on both sides.

Distribute the negative sign to each term in the parenthesis.

Add  and subtract  on both sides.

Divide  on both sides.

We must check each answer. Let's try .

This is true therefore  is a correct answer. Let's next try .

This is not true therefore  is not correct.

### Example Question #3 : How To Find The Solution To An Inequality With Addition

If and , then which of the following could be the value of ?

Explanation:

To solve this problem, add the two equations together:

The only answer choice that satisfies this equation is 0, because 0 is less than 4.

### Example Question #26 : Inequalities

Solve for :

Explanation:

We want to isolate the variable on one side and numbers on another side. Treat like a normal equation.

Subtract  on both sides.

Divide  on both sides.

### Example Question #27 : Inequalities

Solve for :

Explanation:

We want to isolate the variable on one side and numbers on another side. Treat like a normal equation.

We need to set-up two equations since it's absolute value.

Subtract  on both sides.

Distribute the negative sign to each term in the parenthesis.

Add  and subtract  on both sides.

Divide  on both sides.

We must check each answer. Let's try .

This is true therefore  is a correct answer. Let's next try .

This is not true therefore  is not correct.